Oscillations and Multistability in the Electrode Processes of Pseudohalogenide Complexes of Nickel(II) – Experiment and Theory.
(by Marek Orlik and Rafal Jurczakowski)

Marek Orlik

Dynamical self-organization of matter occurs in different physico-chemical systems, including those in which electrochemical processes occur. The recently studied by us systems of this type include the thiocyanate and azide complexes of nickel(II) which undergo electroreduction at mercury electrodes. The electrochemical mechanisms of these processes are responsible for the formation of a single region of the negative differential resistance (NDR) for the nickel(II)-thiocyanate (Ni(II)-SCN-) current-potential (I-E) characteristics, and of one or two NDR regions for the nickel(II)-azide (Ni(II)-N3-) I-E characteristics. It is known that the NDR region, coupled with sufficient serial ohmic resistance, is a source of instabilities leading to spontaneous current oscillations and/or bistability. Particularly interesting is the case of the two consecutive NDR regions which should lead to tristability, a phenomenon easy to predict theoretically, but rarely observed in the real experimental systems.
A short description of the Ni(II)-SCN- electroreduction is useful for the characterization of the experimental procedure applied and the ways of the theoretical analysis performed. The mechanism of formation of the NDR region will be explained. In order to observe true steady-states, sustained oscillations and bistability in this process, the streaming mercury electrode had to be used. Such an electrode, of a special construction, was applied by us for the first time in the studies of self-organization in electrochemical processes [1]. Numerical modeling of bistability [1], as well as the theoretical linear stability analysis of the Ni(II)-SCN- system, related to the specific properties of a streaming electrode, allowed us to construct the stability criteria of the system, including the bifurcation diagram, well concordant with the experimental one [2]. Particular emphasis will be further put on the dynamical characteristics of the azide complexes of nickel(II), the electroreduction of which was studied with the same experimental methodology (streaming electrode) [3]. Under conditions of our experiments no oscillations were reported for the Ni(II)-N3- complexes (due to the slower kinetics of this electrode process, compared to the diffusion transport rate), but instead the bistability and, due to the two NDR regions in the I-E dependence, the tristability were indeed reported (cf. diagram of the stable steady-states, double folded diagram of stable and unstable steady-states) [3]. One should note that there are only few experimental observations of tristability in chemical (including electrochemical) systems. For the understanding of this phenomenon, in each such a case the physico-chemical origin of multiple steady-states should be found. In the particular case of Ni(II)-N3- studied, the electrochemical mechanism of the formation of the sequence of two NDR regions will be explained. It involves the rare case of the electroreduction of Ni(II) central ion, followed by the parallel partial electroreduction of N3- ligand at more negative potentials [4]. In consequence it was possible to construct the model of this process and perform theoretical linear stability analysis of the Ni(II)-N3- electrochemical system. Three dynamical variables were involved in this analysis (the electrode potential and surface concentrations of two electroactive species) and this approach allowed us to construct the bifurcation diagrams for bistability and tristability, well concordant with the experimental ones.
The unstable behavior of such electrochemical processes has common features with the electronic circuits which include e.g. tunnel diodes as the NDR elements. The common way of studying electronic circuits and of the electrochemical processes is the analysis of their impedance response as a function of frequency of ac voltage, at different electric potentials. For the electrochemical dynamical systems, the earlier works of R. de Levie, M. T. M. Koper and J. H. Sluyters indicated criteria of diagnosis of the saddle-node and Hopf bifurcations from the electrochemical impedance spectra. Taking this into account, recently we also performed the electrochemical impedance studies of the above-mentioned processes. Experimental spectra for Ni(II)-SCN- and Ni(II)-N3- electrode processes were measured, particularly for conditions corresponding to electrochemical instabilities (indicating the possibility of the Hopf and saddle-node bifurcations for the Ni(II)-SCN- electroreduction at E = 1.1 V [5]).

The Kramers-Kronig transformation was used to validate the impedance data. During the analysis of the impedance (Z) response we found that the impedance characteristics of a capacitive (i.e. charging the electrode) current at such an electrode is formally similar to that of the faradaic current for the slow process (this means a semicircle in Z(imaginary)-Z(real) space - the Nyquist diagram). This fact was not known before and thus a new theoretical description of the impedance response for our process was necessary, which involved, for the explanation of this capacitive current, an idea of inserting a special virtual resistance into the equivalent electric circuit [6]. The theoretical model which took into account all the specific features of both the electrode type and electrode process studied, was elaborated. The Nyquist diagrams, obtained with this model, appeared to be similar to those reported experimentally for the Ni(I)-SCN- and Ni(II)-N3- systems. These results allowed us to understand the impedance response of the dynamical systems under study.
In our opinion, these experimental and theoretical results constitute a remarkable contribution to the subject of electrochemical instabilities, involving – as the examples - the pseudohalogenide complexes of nickel, studied by dc and ac experimental techniques and analyzed theoretically by the standard techniques of non-linear dynamics. In particular, the reported phenomenon of tristability indicates the ways of searching for the other electrochemical systems of the analogous scheme of the electrode process, in which the electroreduction (or electrooxidation) of the central ion, exhibiting the NDR region, overlaps at more negative (positive) potentials with the partial electroreduction (electrooxidation) of the ligand.